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...rained systems are in the one phase, weakly coupled regime, so that the renormalized potential is still well defined, see [2, 11, 23]. On the other hand examples of non Gibbsianity are given in [16], =-=[17]-=-. Let us now state the main results of [3] on strong Gibbsianity and convergence above Tc in two dimensions. Theorem 1. (Bertini, Cirillo, Olivieri 1999) Consider a two–dimensional Ising system with β...

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...sures. Keywords: Gibbs vs non-Gibbs, generalized Gibbs measures, almost quasilocality, thermodynamic formalism. Mathematical subject classification: Primary: 60G60; Secondary: 82B20, 82B30. 1 Introduction The problem of restoration of Gibbsianness refers to the extension of Gibbsian theory to non-Gibbsian measures observed in statistical mechanics. The latter include measures obtained through renormalization transformations (see [8] and references therein), joint measures of disordered spin systems [19, 20], and measures obtained as a result of stochastic evolutions of Gibbs measures [7]. See [5, 6, 9, 10] for reviews. The goal of the restoration program is the determination of appropriate, more general, classes of measures which would satisfy, in particular, a Gibbsian-like thermodynamic formalism based on the variational principle. Two classes of measures have been introduced so far (1) Weak Gibbsian measures [4, 2, 28]. These are measures that admit an almost sure Boltzman-Gibbs description, that is, whose finite-volume conReceived 7 March 2003. 438 ROBERTO FERNÁNDEZ, ARNAUD LE NY and FRANK REDIG ditional probabilities can be written in terms of an interaction potential which is summable on ...

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...r, Fern'andez and Sokal [8], further examples were found, and a systematic study of Gibbsianness vs. non-Gibbsianness of large classes of transformed or projected versions of Gibbs systems began; see =-=[7, 26, 5, 12, 19, 11, 10, 20, 6, 18]-=- for some of the subsequent work in this area. In particular, Gibbs properties of the so-called fuzzy Potts model were studied in Maes and Vande Velde [26] and H"aggstr"om [18]. Like almost all work i...

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...ally, one ∏ can define a noisy projection onto D ⊂ L through the transformation x∈D exp(p ω′ xωx)/norm. It is well known that renormalization transformations can destroy Gibbsianness (for reviews see =-=[5, 6, 8, 9]-=-). Most of the non-Gibbsian measures resulting from block transformations were shown to be weakly Gibbsian [2, 23]. In Corollaries 3.5 and 3.6 below, we show that in some instances they are, in fact, ...